modeling and simulation of alloys solidification and ... · je tiens à remercier mes incroyables...

POLITECNICO DI MILANOSCHOOL OF INDUSTRIAL AND INFORMATION ENGINEERING

Department of MathematicsMaster of Science in Mathematical Engineering

MODELING AND SIMULATION OFALLOYS SOLIDIFICATION AND

PREDICTION OF SEGREGATIONS

MARIE-ANGE RASENDRAID: 876268

SUPERVISOR: PROF. LORENZO VALDETTARO

INTERNSHIP SUPERVISORS: CHARLES DEMAY

SALMA BELOUAHMARTIN FERRAND

A.Y. 2018-2019

Internship carried out at Electricté de France S.A in the Reserach and Development Departmentat Chatou, France

Acknowledgements

Je remercie grandement mon tuteur de Politecnico, Prof. Valdettaro, qui est toujours disponible

et accessible pour répondre à mes interrogations.

Vorrei ringraziare anche il signore Giovanni Gentili et la signora Michela Gregori per il

sostegno amministrativo preziozo e sempre veloce che mi hanno dato.

Je tiens à remercier mes incroyables tuteurs Salma Belouah, Charles Demay et Martin

Ferrand pour l’aide inestimable qu’ils m’ont apporté tout au long du stage. Ils ont été des

interlocuteurs très complémentaires et j’ai énormèment appris à leurs cotés grace à leurs conseils

et remarques. Je ne saurais assez remercier, plus particulièrement, Charles Demay pour le soutien

inébranlable et la confiance dont il a fait preuve ainsi que pour ses méticuleuses relectures de

mon travail.

Mes plus sincères remerciements à la team développement de Code_Saturne tout spécialement

à Martin Ferrand, Erwan Le Coupanec et Yvan Fournier pour la précieuse aide qu’ils ont apporté.

Un grand merci à toutes les merveilleuses personnes que j’ai rencontré lors de ce stage et

qui participent à créer une ambiance de travail aussi joyeuse. Il serait trop long de citer le nom

de tous les collègues exceptionnels que j’ai eu la chance de cotoyer pendant 6 mois mais je vais

quand meme essayer. Merci à Claire, Thibault, Merlin, Elias, Pablo, Kadra, Thomas, Jerome,

Joseph, Pauline, Paul, Damien, Abdel, Mickael, Roman, Louis, Alvaro, Loic, Christophe, Aurélie

et tous les autres.

Last but not least, gros big up au bureau de l’ambiance alias bureau des stagiaires alias le

bocal. Merci à Camille, Benjamin, Rémi, Mounir, Xuefan, Walid et Azenor.

I

Abstract

This internship falls within a R&D project of Electricité de France which focus on the comprehen-

sion of the causes and the consequences of the positive segregations localized in some identified

components of the French nuclear plants. During ingot castings, a non uniform distribution of the

alloy components can be observed. These heterogeneities at the ingot scale, called macrosegrega-

tions, can impact the mechanical characteristics of the material and in some cases they weaken

its resistance to cracks. In order to progress on the identification and the comprehension of the

main mechanisms leading to macrosegregations, the internship focused on the implementation of

a simple solidification model in Code_Saturne the CFD open-source code developed at EDF. This

work include sensibility studies upon the physical model and upon the numerical methods of a

strongly coupled equations system. The proposed model has been simulated on academic and

industrial cases and compared to other models and to some experimental data.

II

Résumé

Ce stage s’inscrit dans un projet R&D d’EDF, appelé « Gros Forgés », qui se focalise sur la

compréhension des causes et des effets des ségrégations majeures positives présentes sur cer-

tains composants identifiés du parc nucléaire français. Durant la solidification des lingots après

la coulée, il est observé une distribution non uniforme des différents éléments d’alliage. Ces

hétérogénéités à l’échelle du produit, appelées macroségrégations, peuvent impacter les caracté-

ristiques mécaniques du matériau et dans certains cas affaiblir sa résistance aux fissures. Afin

de progresser sur l’identification et la compréhension des mécanismes d’ordre un conduisant

à ces macroségrégations, le stage s’est concentré sur l’implémentation d’un modèle simple de

solidification dans Code_Saturne, le code CFD open-source développé par EDF. Ce travail intègre

des études de sensibilité sur la modélisation physique du phénomène et sur les méthodes de

résolution numérique d’un système d’équations fortement couplées. Le modèle proposé a été mis

en œuvre sur des cas tests académiques et industriels, en se comparant à d’autres modèles de la

littérature et à des données expérimentales.

III

Chi va piano,

va sano e va lontano.

— Italian proverb

Table of Contents

Acknowledgements I

Table of Contents V

Introduction 1

1 Alloy solidification 21.1 Freezing process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Crystals structure map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Enthalpy change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Segregations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Microsegregations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Macrosegregations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Governing equations 112.1 Continuum mixture theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Local conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Interface equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Balance equations for the mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Hypothesis and closure laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.3 Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Listing of the different models simulated . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Numerical modeling 203.1 Code Saturne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

V

TABLE OF CONTENTS

3.2 Simulated model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 SIMPLEC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 PISO-like algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 The improved pressure interpolation in stratified flow . . . . . . . . . . . . . . . . . 25

3.5.1 Continuous formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.2 Application to buoyancy forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Treatment of the porous matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7 Mesh generation and post-processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Validation benchmarks 294.1 Voller and Prakash benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.1 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.2 Geometry of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.3 Characteristics of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.4 Modeling and numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.5 TEST 1: Comparison between two algorithmic schemes . . . . . . . . . . . . 33

4.1.6 Time step sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.7 Mesh sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.8 Numerical results: A comparison with an other CFD software . . . . . . . . 43

4.2 Hebditch and Hunt benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46




4.2.4 TEST 1: Comparison between two algorithmic schemes . . . . . . . . . . . . 49

4.2.5 TEST 2: The drift formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.6 TEST 3: Comparison with and without the modified pressure interpolation 55

4.2.7 TEST 4: Comparison with and without the porosity formulation . . . . . . 57

4.2.8 Time step sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.9 Mesh sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.10 Numerical results: Comparison with SOLID . . . . . . . . . . . . . . . . . . 61

4.2.11 Numerical results: Comparison with 5 other codes . . . . . . . . . . . . . . . 65

4.3 Hebditch and Hunt experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


4.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Industrial application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77



4.4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

VI

TABLE OF CONTENTS

Glossary 83

Appendix A: The incremental PISO-like formulation 83

Appendix B: The averaged equations model 85

Bibliography 87

VII

Introduction

Ingot casting is a manufacturing process used in many industrial sectors such as in the naval,

the aeronautic and the nuclear sectors. During solidification some chemical heterogeneities called

segregations can appear at the ingot scale. Zones of the alloy that are enriched in solute, compared

to the initial concentration, are called positive segregations. On the contrary, depleted zones are

called negative segregations. This phenomena is well-known by metallurgists who handle it by

removing the highly carbon concentrated part of the ingot at the end of solidification. Indeed,

these defaults could not been rectified once the ingot is entirely solidified, so highly positive

segregations are cut off the ingot. This matter is all the more concerning that segregations inside

a metal can lessen its resistance and increase the propagation of cracks. That is why there exists

a necessity to understand better the emergence and the development of segregations during the

freezing process.

Electricité de France (EDF) is a French electric utility company and a global leader in low-

carbon energy. EDF uses steel ingots casted by other companies to forge many components of its

plants and has been exposed to segregations issues. Indeed in 2014, they noticed several defects

on some reactor vessel components of the Flamanville 3 nuclear plant. Then they discovered

that 46 steam generators of the French nuclear plants could be concerned by this issue. This

defect is made by a too high concentration of carbon inside the metal composing the bottom of

the steam generator. Such a discrepancy, called a positive carbon segregation, can weaken the

finished product. This master thesis falls within EDF’s will to understand better the development

of segregations during ingot castings.

In Chapter 1, we focus on the thermodynamical principles behind solidification. In Chapter

2, we derive a mathematical model describing the freezing process based on the continuum

mixture theory and on some set of hypothesis. In Chapter 3, the implementation of the model

on Code_Saturne is presented. We introduce a numerical scheme to treat better the strong

coupling between the set of equations derived in Chapter 2. Besides some numerical artifacts are

introduced to handle better with the pressure interpolation and with the porosity of the alloy. In

Chapter 4, we validate the code thanks to well-known benchmarks and some experimental data.

Finally, we simulate an industrial ingot casting case.

1

Alloy solidication1D uring alloy solidification, some chemical heterogeneities can emerge at the macroscopic

scale of the ingot. These defaults created during the freezing process are irreversible

and have important repercussions on the physical properties of the final material. In

the first part, the main solidification principles are presented. We focus on the different equilibrium

states of a solidifying alloy by studying its phase diagram and by recalling some fundamental

thermodynamics notions. Then we investigate the freezing process at a microscopic scale and the

emergence of a typical crystallized microstructure. This crystallized matrix inside the mushy zone

create local heterogeneities in concentration called microsegregations. Finally, we present four

physical phenomena that transport the microsegregations to form bigger chemical heterogeneities

called macrosegregations.

1.1 Freezing process

1.1.1 Phase diagram

Freezing is a phase change from liquid to solid observed when a mixture reaches a temperature

below its melting or freezing temperature. Unlike pure metals, an alloy does not solidify at a

constant temperature, instead it freezes inside a solidification temperature interval bounded by

the solidus and the liquidus temperature. Above the liquidus temperature Tl the alloy is entirely

liquid, below the solidus temperature Ts it is entirely solid. Inside this interval, the liquid and

the solid phases of each component of the alloy are coexisting, this is called the mushy zone. In

this zone, the solid phase forms a porous matrix through which the residual liquid flows.

In this work, we consider only binary alloys in order to simplify the model and we derive

thermodynamical properties from its phase diagram. A phase diagram is indeed a representation

of the different thermodynamics variables combinations under which each phase exists at

equilibrium. It is a tool to understand better the different thermodynamical mechanisms at stake

during a freezing process.

Below, we study the phase diagram of a binary alloy of components A and B and we describe

2

1.1. FREEZING PROCESS

the different phases that the alloy experiences when freezing.

Figure 1.1: Phase diagram of a binary alloy of components A and B. The liquidus line, in red,is the border between the liquid and the mushy zones. The solidus line, in blue, is the borderbetween the solid and the mushy zones.

1. Liquid zone.

The alloy considered has a concentration C0 of solute B. Initially the temperature is set to

T0 and the alloy is entirely liquid.

2. Mushy zone.

Decreasing the temperature, we hit the liquidus line. At this point, one crystal of solid

forms and we enter into the mushy zone. In this 2-phase zone, at a given temperature T,

the liquid phase has a concentration of solute Cl higher than that of the solid Cs. This is

due to a higher solubility of the solute inside the liquid than in the solid phase. So that a

solidifying solution rejects solute inside the surrounding liquid which increases the local

concentration of the liquid Cl and decreases that of the solid Cs. Thus during a freezing

process, there is a redistribution of solute at the interface liquid-solid.

This creates a chemical heterogeneity called microsegregation that can be quantified thanks

to the partition coefficient kp.

kp = C∗s

C∗l

(1.1)

3


The superscript ∗ means that the concentration is evaluated at the interface. We can notice

that kp is always smaller than unity.

If kp = 1 there is no local redistribution of solute, the concentrations are the same in

both phases. On the opposite, the more kp tends to 0, the more we will observe chemical

heterogeneities at the interface liquid-solid.

We perform a solute balance at the interface liquid-solid, recalling that, in a closed system,

the initial concentration of solute C0 is conserved during the freezing process.

C0 = f lC∗l + fsC∗

s (1.2)

with f l (resp. fs) the mass fraction of solute inside the liquid phase (resp. solid phase).

3. Solid zone.

Going on cooling we will hit the solidus line if C0 < Cs,eut or the eutectic level if C0 ≥ Cs,eut.

In both cases the last atom of liquid will solidify at this point and we will end up with a

completely solid alloy.

In the reality the liquidus and the solidus borders are not straight lines, but we assume that

they follow the following linear relations.

T = Tmelt +mlCl (1.3)

T =

Tmelt +

ml

kpCs i f C0 < Cs,eut

Teut i f C0 ≥ Cs,eut

(1.4)

with ml the liquidus slope.

An equivalent formulation is

Tl = Tmelt +mlC0 (1.5)

Ts =

Tmelt +

ml

kpC0 i f C0 < Cs,eut

Teut i f C0 ≥ Cs,eut

(1.6)

1.1.2 Crystals structure map

When an alloy reaches the liquidus line, a grain of solid appears, it is called the nucleation point.

Several nucleation points can be at different places of the ingot. Then atoms of liquid gather

around nucleation points and bond together in a way to form crystals.

Often the shape of the crystal will be similar to a Christmas tree called dendrites and sketched

in Figure (1.2). During the freezing of a large ingot, many dendrites form and grow until they

encroach upon other dendrites. If some residual atoms of liquid are trapped between the dendrites,

they may solidify to form a more regular crystal. Otherwise there is not enough atoms to fill the

interdendritic space and a void is left.

4


λ1

λ2

Figure 1.2: Dendrites with the Primary Dendrite Arm Spacing λ1 and the Secondary DendriteArm Spacing λ2

Two main numbers, represented in Figure (1.2), characterize the dendrites: the primary

dendrite arm spacing λ1 and the secondary dendrite arm spacing λ2.

During ingot casting, three solidification layers represented in Figure (1.3) can be observed:

• The outer equiaxed zone. It is a thin layer covering the mold. When the hot liquid alloy is

poured into the mold, the first atoms in contact with the mold will instantaneously freeze.

This forms a first solidified layer of packed equiaxed grains.

• The columnar zone. Unlike in the outer equiaxed zone, the crystals formed in the columnar

zone grow in a preferential direction given by the temperature gradient. In this zone the

solidification takes more time and the crystals often grow following a dendritic pattern.

• The inner equiaxed zone. It is composed by the surrounding liquid and by moving dendritic

grains coming from the dendrites of the columnar zone or from the nucleation points.

5


Figure 1.3: Structure of the crystals during a casting process. Image taken from [4]

1.1.3 Enthalpy change

The enthalpy h is often used to describe energy transfer especially in the case of phase change

problems. It is a thermodynamic quantity linked to the internal energy U, the pressure p and

the volume V of the system by the following relation.

h =U + pV (1.7)

In this work all the simulated systems were closed and the freezing was processing at a fixed

pressure. For this reason the variation of enthalpy equals the heat released or absorbed. Solidifi-

cation is an exothermic process, some thermal energy, called latent heat of fusion L, is discharged.

It represents the heat energy released by the system during its transformation from liquid to

solid. That is why the total variation of enthalpy during freezing writes:

∆h =−L (1.8)

Besides for processes at constant pressure, we can demonstrate that the enthalpy simplifies in:

dh = cpdT (1.9)

with cp the specific heat at constant pressure of the system.

We can define enthalpies prevailing inside each isolated phases so that hl stands for the

enthalpy inside the all-liquid phase and hs for the enthalpy inside the all-solid phase. From

6

1.2. SEGREGATIONS

equation (1.8) and (1.9), we deduce: hl = cp,lTl +L

hs = cp,sTs(1.10)

1.2 Segregations

A segregation is an heterogeneity of the chemical composition inside an ingot. Three scales of

segregations can be observed at the end of the freezing process.

• Microsegregations (∼ 100µm)

• Mesosegregations (∼ cm)

• Macrosegregations (∼ m)

We are interested in the macro and the mesosegragtions because they can weaken the mechanical

properties of the manufactured piece. Yet, we will also study the microsegregations because they

cause the segregations observed at bigger scales.

1.2.1 Microsegregations

Microsegregations are the chemical inhomogeneities observed at a microscopic scale due to the

solute redistribution at the interface solid-liquid. This phenomenon has been presented in section

(1.1.1) and is the cause of all the other segregations.

In a neighborhood of the interface solid-liquid, we should define a microsegregation model

which gives a relation between the concentration Ck in the phase k and the concentration C∗k

in the phase k at the interface. With k = l in the liquid phase and k = s in the solid phase. The

following two simple microsegregation models are often used in the literature:

• The lever rule.

It supposes a perfect diffusion of solute in both liquid and solid phases. So that we have

Cs = C∗s and Cl = C∗

l (1.11)

• The Gulliver-Scheil rule.

It supposes a perfect diffusion of solute in the liquid phase and no diffusion in the solid

phase. It means that

Cs 6= C∗s and Cl = C∗

l (1.12)

In the following figure, we represented the lever rule and the Gulliver-Scheil rule at the

interface liquid-solid of a dendrite.

7

1.2. SEGREGATIONS

Figure 1.4: Two usual microsegregation models at the interface solid-liquid of a dendrite. Left:Lever rule. Right: Gulliver-Scheil rule.

1.2.2 Macrosegregations

In this work we want to predict the development of macro and mesosegregations. They are identi-

fied according to their metallurgical appearance. There are the so-called A and V segregations,

the negative cone segregation and the positive hot top segregation.

8

1.2. SEGREGATIONS

Figure 1.5: Usual segregations shapes inside a solidified ingot

Macro and mesosegregations are caused by microsegregations. Indeed, the chemical het-

erogeneities at the micro scale are then transported inside the whole ingot creating bigger

segregations at localized parts. Several phenomena are responsible for transporting microsegre-

gations [6]:

Figure 1.6: Main phenomena of solute transport

9

1.2. SEGREGATIONS

(a) Thermosolutal convection.

The alloy is subjected to important thermal gradients due to its cooling from the mold.

There are also solutal gradients from the chemical heterogenities at the interface and their

displacements towards localized part of the ingot. These high gradients create convection

movements that can be in the same or in opposite directions.

(b) Grain movements.

Grains in movement inside the mushy zone are coming from nucleation points or from

dendritic crystals detached from the columnar zone. These grains gather in the bottom of the

ingot by gravity which is one of the cause of the negative base segregation. Indeed we recall

that the concentration of solute inside the solid is smaller than the initial concentration C0.

(c) Shrinkage.

It is the contraction of the material during its cooling due to a higher density of the solidified

material comapared to its liquid state. While contracting, the porous solid suck up the

surrounding liquid.

(d) Deformation of the solid.

Under stresses, the porous matrix behaves like a sponge and it ejects concentrated liquid

outside of the solid.

10

Governing equations2T he major difficulty to model solidification lies in the treatment of the so-called mushy

zone. This region contains a mixture of both liquid and solid components. The math-

ematical models for solving phase change problems subject to natural convection can

be divided roughly into two sections. The first one is based on a multiphase approach where the

equations are solved in each phase and coupled with appropriate boundary conditions at the

phase interface. This requires a front interface tracking technique and to update continuously

the two domains. The technique is often applied to pure substances because they present a sharp

interface between liquid and solid phases. On the contrary, the phase change of multicomponents

alloys occurs during a large range of temperature where a permeable solid matrix coexist with

the liquid phase. In this case, a second technique based on a single region modeling seems more

suitable. It consists of solving a unique system of momentum, heat and species transport equations

in the whole domain composed by solid, mushy and liquid zones. The interface contributions are

taken into account by adding appropriate source terms. An advantage is that a single fixed grid is

needed.

In this work, we focused on the second approach and more particularly on the continuum mixture

model that we derive following a similar approach as in [12]. This model relies on volume averag-

ing techniques based on the classical mixture theory. The averaged-equations model is another

one-region model that is also widely used in the literature and is presented in Appendix B.

2.1 Continuum mixture theory

Mixture theory provides a continuum framework to model multiphases and multicomponents

systems into single domain systems. It postulates that at any instant of time each material point

of a body is occupied by a finite number of particles, one for each component of the mixture.

In this way the mushy zone of a freezing alloy can be seen as a superposition of continuously

distributed phases and of interpenetrable continua.

11

2.2. LOCAL CONSERVATION LAWS

2.2 Local conservation laws

We infer the conservation laws ruling our system following the same steps as in [12]. Using an

Eulerian description, we consider a fixed control volume. It should be smaller than the ingot

scale, but large enough to include the main features of the microstructure. Its dimensions are in

the mesoscale.

Solidifiedgrains

Surroundingliquid

Mesoscale

Microscale

Figure 2.1: Representative Elementary Volume

We represent in Figure (3.1) a representative elementary volume (REV) of volume V and of

surface area A located inside the mushy zone. The REV contains solid components of total volume

Vs and liquid components of total volume Vl so that we have

V =Vs +Vl (2.1)

We define the solid volume fraction gs and the liquid volume fraction gl by

gs = Vs

Vand gl =

Vl

V(2.2)

From equation (2.1), we can deduce a relation between the two volume fractions.

gs + gl = 1 (2.3)

We denote by k one of the two isolated phase, so that k = l in the liquid phase and k = s in the

solid phase. We write the balance equation of a scalar quantity φk specific to phase k inside a

fixed control volume of volume V.

∂t

∫V

(ρkφk)dVk +∫

A(ρkφkuk) ·n dAk =−

∫A

Jk ·n dAk +∫

VSk dVk (2.4)

12

2.2. LOCAL CONSERVATION LAWS

with Ak = gk A, Jk a surface flux vector and Sk a volumetric source term. Applying the Gauss the-

orem and recalling that dVk = gk dV and dAk = gk dA, we end up with the following differential

formulation.

∂t(gkρkφk)+div(gkρkφkuk)=−div(gk Jk)+ gk Sk (2.5)

• Continuity equation.

The mass conservation can be derived from equation (2.5) by taking φk = 1, Jk = 0 and

Sk = Mk.∂gkρk

∂t+div(gkρkuk)= gkMk (2.6)

with Mk represents a rate of mass production or annihilation inside phase k.

• Momentum conservation equation.

The momentum equation is deduced from equation (2.5) by taking φk = uk,i, Jk =−Ck,i and

Sk = ρk g i + Gk,i.

Having denoted by (e1, e2, e3) a basis of our problem and by i one of the three coordinates

of the basis, we define:

uk,i = uk · e i with i = 1, 2, 3

Gk,i = Gk · e i

g i = g · e i

Ck,i = Ck · e i

Gk represents the production or annihilation of momentum due to phase interactions such

as drag and lift. Ck denotes the Cauchy stress tensor which can be decomposed into two

parts: a pressure component and an extra stress term τk

Ck =−p I +τk (2.7)

So that we have for coordinate i:∂(gkρkuk,i)

∂t+div(gkρkuk,iuk)= div(gkCk,i)+ gkρk g i + gkGk,i

The local momentum equation can then be written in a vectorial form.

∂(gkρkuk)

∂t+div(gkρkuk ⊗uk)=−∇(gk pk)+div(gkτk)+ gkρk g (2.8)

• Energy conservation equation.

The heat equation is deduced from equation (2.5) by taking φk = hk, Jk = −λ∇T and

Sk = Ek.∂(gkρkhk)

∂t+div(gkρkhkuk)= div(gkλk∇Tk)+ gkEk (2.9)

with λk the thermal conductivity and Ek a rate of energy production or annihilation inside

phase k.

13

2.3. INTERFACE EQUATIONS

• Solute transport equation.

The heat equation is deduced from equation (2.5) by taking φk = Ck, Jk = −Dk∇Ck and

Sk = Sk.∂(gkρkCk)

∂t+div(gkρkCkuk)= div(gkρkDk∇Ck)+ gkSk (2.10)

With Dk the diffusive coefficient in phase k and Sk the rate of production or annihilation of

solute inside phase k.

2.3 Interface equations

Mk, Gk, Ek and Sk represent the physical transfers between the liquid and solid phases due

to chemical reactions, interaction forces and energy transfer. We make the hypothesis that the

mixture as a whole is a closed-system, so that the production and annihilation terms are only

caused by exchanges among constituents and phases. In other words, the whole mixture should

behave like a single body. This assumption is expressed by the following relations:

gsMs + gl Ml = 0 (2.11)

gsGs + glG l = F (2.12)

gsEs + gl E l = 0 (2.13)

gsSs + gl Sl = 0 (2.14)

where F represents the force due to phase interactions.

2.4 Balance equations for the mixture

2.4.1 General formulation

To derive the continuum balance equations for the mixture conservation, we introduce the

mixture density ρm, the mixture pressure pm and the mass averaged velocity um, concentration

Cm and enthalpy hm.

ρm = gsρs + glρl (2.15)

pm = gs ps + gl pl (2.16)

um = 1ρm

(gsρsus + glρl ul) (2.17)

Cm = 1ρm

(gsρsCs + glρlCl) (2.18)

hm = 1ρm

(gsρshs + glρl hl) (2.19)

14

2.4. BALANCE EQUATIONS FOR THE MIXTURE

Then we sum up the two microscopic equations specific to each phase and we use the interface

relations. So that we end up with the following system.

∂(ρmum)

∂t+div(glρl ul ⊗ul + gsρsus ⊗us)=−∇(pm)+div(glτl + gsτs)+ρm g+F

∂(ρmhm)∂t

+div(glρl hl ul + gsρshsus)= div(glλl∇Tl + gsλs∇Ts)

∂(ρmCm)∂t

+div(glρlCl ul + gsρsCsus)= div(glρlDl∇Cl + gsρsDs∇Cs)

∂ρm

∂t+div(ρmum)= 0

(2.20)

2.4.2 Hypothesis and closure laws

To simplify the system of equations (2.20), we make the following hypothesis.

1. Thermodynamic equilibriaWe consider that

• There is a local thermal equilibrium inside the control volume.

Ts = Tl = T (2.21)

• The specific heat cp, the dynamic viscosity µ, the diffusion coefficient D and the

thermal conductivity λ are constant and equal in both phases.

cps = cpl = cp (2.22)

µs =µl =µm =µ (2.23)

Ds = Dl = D (2.24)

λs =λl =λ (2.25)

ρs = ρl = ρ (2.26)

Remark 1. These approximations are widely used as a first approximation of the model

[10], [9], [7]. The future work of this thesis will consider different values of the coefficients

listed above inside the liquid and the solid phase.

From equation (2.26), we deduce that the mass fraction in phase k, fk, simplifies as

fk =ρk

ρgk = gk (2.27)

Besides, from equations (2.22) and (1.10), we can write the mixture enthalpy as follows

hm = gshs + gl hl = cpT + glL (2.28)

15


2. Newtonian fluidWe suppose the fluid to be Newtonian inside both phases. The solid phase being modeled

as a highly viscous Newtonian fluid. Then the stress tensor is defined by the constitutive

relation

τ= 2µD+λ(trD)I (2.29)

with µ the shear viscosity, λ the dilatational viscosity and D = 12 (∇u+(∇u)T ) the deformation

rate tensor.

Using the continuity equation, we have tr(D)= 0 so that τ= 2µD.

3. The Boussinesq approximationThe Boussinesq approximation applied to the mixture consists of supposing little changes

of the density compared to its mean value ρre f . It states that ρm is constant (ρm = ρl = ρs =ρre f ) except in the buoyancy term of the momentum equation where it is equals to ρb and

where it varies linearly with the temperature and the solute mass concentration in liquid.

ρb = ρre f (1−βT (T −Tre f )−βc(Cl −Cre f )) (2.30)

4. Darcy law in the mushy zoneWe suppose that the flow inside the mushy zone follows a Darcy law. This is a usual

description for modeling a porous flow. It is valid for laminar flow through sediments or

through dendrites interstices in our case. It states that

u =− Kµl

∇p (2.31)

K denotes the permeability, it is function of the porosity gl . Then the drag force F modeling

the porous flow inside the mushy zone can be written as follows.

F =−µl

Kum (2.32)

The permeability of the porous matrix is often modeled by the Karman-Cozeny law.

K = λ22

180

g3l

(1− gl)2 (2.33)

with λ2 presented in Figure (1.2).

5. Motionless solid phase (Columnar zone assumption)We suppose that the mushy region can be modeled as a columnar zone. In this case we drop

off nucleation and grain movements phenomena and we consider that the solid matrix is

fixed so that we have

us = 0 (2.34)

Then the mixture velocity um reduces to the superficial velocity um = gl ul .

16


6. The lever ruleThe lever rule, presented in section (1.2.1), is used as microsegregation model. It supposes

that there is a perfect diffusion inside both liquid and solid phases so that we have

Cs = kpCl (2.35)

From equation (2.27) and applying the lever rule, the solute balance at the interface, written

in equation (1.2), simplifies in

C0 = glCl + gsCs (2.36)

We recognize the definition of the mixture concentration Cm. We could expect such a result

because we are now performing a solute conservation at the interface with an initial

concentration inside the control volume equals to Cm. In other words what we called C0

in the first chapter has been called Cm when applied to a representative volume element.

Thus, using equation (2.3), we obtain

gs = Cl −Cm

Cl −Csand gl =

Cm −Cs

Cl −Cs(2.37)

7. Closure laws for the volume fractionsIn the whole liquid phase we set gl = 1, on the contrary in the solid phase gl = 0. In the

mushy zone however we need a further study to derive the liquid volume fraction equation.

We suppose that the liquidus and the solidus curves can be approximated by a straight line.

a) If Cm ≥ Cs,eut

The closure law for gl (2.37) writes

gl =Cm −Cs,eut

Ceut −Cs,eut(2.38)

b) If Cm < Cs,eut

From equations (1.3) and (1.5), we have

Cm = 1ml

(Tl −Tmelt)

Cl =1

ml(T −Tmelt)

Cs =kp

ml(T −Tmelt)

Injecting these equation inside (2.37) we end up with

gl = 1− 11−kp

T −Tl

T −T f(2.39)

Remark 2. It should be notice that the closure law for gl depends on the chosen

microsegregation model.

17

2.5. LISTING OF THE DIFFERENT MODELS SIMULATED

2.4.3 Balance equations

In this section, we rewrite the balance equations (2.20) using the previous hypothesis. For

the sake of clarity, we will drop the m subscript notation for all the mixture variables, but

it remains implied.

∂(ρu)∂t

+div(glρl ul ⊗ul)=−∇(p)+div(µ∇u)+ρb g− µl

Ku

div(u)= 0

∂(ρh)∂t

+div(glρl hl ul)= div(λ∇T)

∂(ρC)∂t

+div(glρlCl ul)= div(ρD∇C)

(2.40)

which is equivalent to

∂(ρu)∂t

+div(ρu⊗u)=−∇(p)+div(µ∇u)+ρb g− µl

Ku−div(Σu)

div(u)= 0

∂(ρh)∂t

+div(ρhu)= div(λ∇T)−div(Σh)

∂(ρC)∂t

+div(ρCu)= div(ρD∇C)−div(ΣC)

(2.41)

Σu, Σh and ΣC are crossed-terms defined by

Σu = glρul ⊗ul −ρu⊗u

Σh = glρhl ul −ρhu

ΣC = glρlCl ul −ρCu

2.5 Listing of the different models simulated

During our simulations we tested different ways to express the crossed-terms Σu, Σh and ΣC .

The numerical results of the models presented below can be found in chapter 4.

• Usually, the inertial contribution Σu is omitted because its action is negligible compared to

the Darcy damping force F.

Σu = 0 (2.42)

Remark 3. All the same, we also tested the case for which Σu is non null in our simulations,

but did not notice significant changes of the results.

• Σh is considered to be null in [10] so that the enthalpy equation becomes

∂(ρh)∂t

+div(ρhu)= div(λ∇T) (2.43)

18

2.5. LISTING OF THE DIFFERENT MODELS SIMULATED

Using equation (2.28) we can rewrite 2.43 with the temperature as unknown.

∂(ρcpT)∂t

+div(ρcpTu)= div(λ∇T)−Sh (2.44)

with Sh = ∂(ρgl L)∂t +div(ρglLu).

This formulation is widely used among continuum mixture theory models, but we chose to

take Σh into consideration.

Σh = glρhl ul −ρhu = ρ(cpT +L)u−ρ(cpT + glL)u

= ρL(1− gl)u

So that we have the following enthalpy equation

∂(ρh)∂t

+div(ρhu)= div(λ∇T)+div(ρLgl u) (2.45)

Once again we deduce the equivalent formulation in temperature

∂(ρcpT)∂t

+div(ρcpTu)= div(λ∇T)−S′h (2.46)

with S′h = ∂(ρgl L)

∂t

• ΣC simplifies as

ΣC = glρCl ul −ρCu = ρ(Cl −C)u (2.47)

A first raw implementation was to treat ΣC as a source term of the right hand side, so that

we have the following solute transport equation.

∂(ρC)∂t

+div(ρCu)= div(ρD∇C)−div(ρ(Cl −C)u) (2.48)

A second implementation was to include ΣC inside the drift term in order to treat it

implicitly.

From equation (2.37), we can derive the following relation between C and Cl .

C = Cl(kp + gl(1−kp))

Injecting this relation into equation (2.47), we obtain the following drift formulation.

∂(ρC)∂t

+div(ρηCu)= div(ρD∇C) (2.49)

with η defined as follows:

η= 1gl + (1− gl)∗kp

(2.50)

19

Numerical modeling3T his chapter presents the numerical methods tested to simulated our model inside

Code_Saturne, an open-source computational fluid dynamics software. Commonly, to

solve the Navier-Stokes equations the SIMPLEC (Semi-Implicit Method for Pressure

Linked Equations - Consistent) algorithm is used. Yet, in our case, the mathematical model derived

in chapter 2 is subject to a strong coupling between its different conservations equations. Thus, we

investigated an other algorithm, called PISO-like, to treat better this strong coupling.

Besides, we added two numerical artifacts to improve the pressure interpolation and the treatment

of the porosity in the mushy zone.

3.1 Code Saturne

Code_Saturne is an open-source computational fluid dynamics software developed at Electricité

de France R&D [1]. It is used for industrial applications and research activities in several

fields related to energy production, process engineering (plasma, electric arcs), gas and coal

combustion, turbomachinery, aeraulics (ventilation, pollutant dispersion). It solves the Navier-

Stokes equations for 2D, 2D axisymmetric and 3D flows in steady or unsteady, laminar or

turbulent, dilatable, incompressible, isothermal cases. The solver is based on a co-located finite

volume approach for the space discretization, we plus imposed an upwind scheme. For the time

discretization, we set an implicit Euler scheme.

20

3.2. SIMULATED MODEL

Figure 3.1: Numerical simulations on Code_Saturne

Basic physical problems can be implemented on Code_Saturne via the grahical user interface

(GUI). But to implement our solidification problem, we mainly used user subroutines written in

Fortran90 or in C and added to the core code of Code_Saturne.

3.2 Simulated model

We consider the mixture model (3.1) and we add equations (2.42), (2.46) and (2.47) for the

treatment of the crossed terms. In this way we obtain the following system of 4 equations.

∂(ρu)∂t

+div(ρu⊗u)=−∇(p)+div(µ∇u)+ρb(T,Cl)g− µ

K(gl(T,C))u

div(ρu)= 0

∂(ρcpT)∂t

+div(ρcpTu)= div(λ∇T)− ∂(ρgl(T,C)L)∂t

∂(ρC)∂t

+div(ρCu)= div(ρD∇C)−div(ρ(Cl −C)u)

(3.1)

The total number of unknowns of our system is 8: (u, p,T, gs, gl ,Cs,Cl ,C).

21

3.3. SIMPLEC ALGORITHM

Thus we need to add 4 other equations to obtain a closed system.

gs + gl = 1

C = gsCs + glCl

Cs = kpCl

Tl = Tm +mlCl

(3.2)

Remark 4. From now on and for the sake of clarity, we will stop writing the dependence of gs, gl

and K on temperature and concentration, but it remains underlying.

With this convention, we will have the following equivalence for a quantity ψ

ψ≡ψ(T,C) ψn ≡ψ(Tn,Cn) ψn,k ≡ψ(Tn,k,Cn,k)

3.3 SIMPLEC algorithm

SIMPLEC is a widely used algorithm to solve the Navier-Stokes equations. The velocity-pressure

coupling is handled by performing the usual prediction and correction steps. We solve the scalar

equations in T and C before the mass and momentum equations.

The variables u, p,T and C are computed at the center of the cell. We denote by Q the mass flux

Q = ρu, it is calculated at the face. At time n we perform the following steps.

i Solve scalar equationsThe first step is to solve the scalar equations using a backward Euler method in time.

ρCn+1 −Cn

∆t+div(Cn+1 Qn)=−div((Cn

l −Cn+1)Qn)+div(ρD∇Cn+1) (3.3)

ρcpTn+1 −Tn

∆t+ cp div(Tn+1 Qn)= div(λ∇Tn+1)−

∆(ρgn/n+1l L)

∆t(3.4)

Remark 5. The notation n/n+1 in the transient term ∆(ρgn/n+1l L) of the heat equation

means that it has been decomposed into two parts one explicit and one implicit. Indeed the

chain rule gives.∂(ρglL)

∂t= ∂(ρglL)

∂T∂(T)∂t

+ ∂(ρglL)∂C

∂(C)∂t

(3.5)

Discretizing (3.4) over time we write

∆(ρgn/n+1l L)

∆t= ∂(ρglL)

∂T

∣∣∣∣n

Tn+1 −Tn

∆t+ ∂(ρglL)

∂C

∣∣∣∣n

Cn+1 −Cn

∆t(3.6)

ii Compute the closure laws

22

3.4. PISO-LIKE ALGORITHM

iii Velocity predictionThe prediction step solves the momentum equation with an explicit pressure pn and returns

a predicted velocity u.

ρu−un

∆t+div(u⊗Qn)= div(µ∇ u)−∇pn− µ

Kn u+ρ[1−βT (Tn−Tre f )−βC(Cnl −Cre f )]g (3.7)

iv Pressure correctionThe predicted velocity u does not respect the continuity equation a priori. In this step,

we correct the pressure adding a scalar function φ to pn to make sure that the computed

velocity field is divergence free. ρ

un+1−u∆t =−∇φ (∗)

div(Qn+1)= 0(3.8)

Applying the divergence operator to (∗) and using the continuity equation, we find the

following Poisson problem.

div(∆t∇φ)= div(ρ u) (3.9)

v Pressure and velocity updateNext, we update the pressure and velocity thanks to the scalar function φ found by solving

the former Poisson problem. It yields:

pn+1 = pn +φ (3.10)

Using (∗), we deduce.

un+1 = u− ∆tρ

∇φ (3.11)

Remark 6. Scalar equations might be solved after the velocity-pressure coupled equations but it

has no influence in the framework of the SIMPLEC algorithm.

3.4 PISO-like algorithm

The PISO-like algorithm uses the same steps as SIMPLEC except that some sub-iterations are

performed at each time step in order to treat better the strong coupling between the scalar

equations and the Navier-Stokes equations.

This version solves the scalar equations before tackling the prediction and correction steps in

order to have an accurate estimation of the buoyancy force.

23

3.4. PISO-LIKE ALGORITHM

At time n, we perform the following initialization.

un+1,0 = un

pn+1,0 = pn

Tn+1,0 = Tn

Cn+1,0 = Cn

Qn+1,0 =Qn

At the k− th sub-iteration, we have

i Solve scalar equations

ρCn+1,k −Cn

∆t+div(Cn+1,k Qn+1,k−1)=−div((Cn+1,k−1

l −Cn+1,k)Qn+1,k−1)+div(ρD∇Cn+1,k)

(3.12)

ρcpTn+1,k −Tn

∆t+ cp div(Tn+1,k Qn+1,k−1)= div(λ∇Tn+1,k)−

∆(ρgn+1,k−1/kl L)

∆t(3.13)

Repeating the same steps as in Remark 2, we have

∆(ρgn+1,k−1/kl L)

∆t= ∂(ρglL)

∂T

∣∣∣∣n+1,k−1

Tn+1,k −Tn+1,k−1

∆t+ ∂(ρglL)

∂C

∣∣∣∣n+1,k−1

Cn+1,k −Cn+1,k−1

∆t(3.14)

ii Compute the closure laws

iii Velocity prediction

ρuk −un

∆t+div(uk ⊗Qn+1,k−1)= div(µ∇ uk)−∇pn+1,k−1 − µ

Kn+1,k uk

+ρ[1−βT (Tn+1,k −Tre f )−βC(Cn+1,kl −Cre f )]g (3.15)

iv Pressure correction ρ

un+1,k−uk

∆t =−∇φk

div(Qn+1,k)= 0(3.16)

Applying the same method than previously we find the following Poisson problem.

div(∆t∇φk)= div(ρ uk) (3.17)

v Pressure and velocity update

pn+1,k = pn+1,k−1 +φk (3.18)

un+1,k = uk − ∆tρ

∇φk (3.19)

24

3.5. THE IMPROVED PRESSURE INTERPOLATION IN STRATIFIED FLOW

vi Stopping criteriaWe stop doing the PISO-like sub-iterations when

||un+1,k −un+1,k−1||L2 ≤ ε||un||L2 (3.20)

with ε= 10−5 by default.

Then we move to the next time step n+1, by introducing

un+1 = un+1,k

pn+1 = pn+1,k

Tn+1 = Tn+1,k

Cn+1 = Cn+1,k

Qn+1 =Qn+1,k

Remark 7. The second version we have tested consists in solving the scalar equations after the

treatment of the Navier-Stokes equation. However, regarding the strong coupling between (T,C)

and u through the source terms, the proposed version seems better.

Remark 8. We could also perform the sub iterations only on the Navier-Stokes equation and solve

the scalar equations outside of the PISO loop. But this method is not relevant when considering

driving source terms such as the buoyancy force.

3.5 The improved pressure interpolation in stratified flow

3.5.1 Continuous formulation

The improved pressure interpolation in stratified flow (iphydr = 1 in Code_Saturne) is an algo-

rithmic modification to improve the treatment of the external forces.

We decompose the pressure into two parts the hydrostatic pressure Ph and the dynamic

pressure P ′.P = Ph +P ′ (3.21)

By definition the hydrostatic component is the pressure in equilibrium with the external forces.

More precisely, it is a solution of the following balance ∇Ph = f .

Let’s consider the following momentum equation

∂udt

+ρ(u ·∇)u = div(µ∇u)−∇P + f +Su (3.22)

f represents the external forces which are independant from the velocity such as the gravity, the

electrical force, the centrifugal force etc and Su stands for all the other external forces.

25

3.5. THE IMPROVED PRESSURE INTERPOLATION IN STRATIFIED FLOW

Splitting the pressure into its dynamic and hydrostatic components, (3.22) simplifies

∂udt

+ρ(u ·∇)u = div(µ∇u)−∇P ′+Su (3.23)

When we solve the momentum equation by a finite volume method, we need to have access to

the value of the pressure at each faces of the element.

Indeed from the divergence theorem

∫Ω∇P ′dΩ= ∑

f acesP ′

f aces S f aces

So we need to perform an interpolation using the values of pressure computing at the center

of each cell. We make the interpolation only on the dynamical component of the pressure, because

the gradient of the hydrostatic component cancel with the external forces.

Figure 3.2: Cells configuration

Let’s take α ∈ [0,1],

P ′F =αP ′

I + (1−α)P ′J (3.24)

PF = PhF +P ′

F (3.25)

= PhF +αP ′

I + (1−α)P ′J (3.26)

= PhF +αPI + (1−α)PJ −αPh

I − (1−α)PhJ (3.27)

Since f =∇Ph, we have

PhI = Ph

F +FI · f I (3.28)

PhJ = Ph

F +F J · fJ (3.29)

Thus

PF =αPI + (1−α)PJ −αFI · f I − (1−α)F J · fJ (3.30)

We recover the usual interpolation formula for the total pressure supplemented by some corrective

terms.

26

3.6. TREATMENT OF THE POROUS MATRIX

3.5.2 Application to buoyancy forces

The PISO-like algorithm is subjected to little changes when we use equation (3.30) to better

interpolate the pressure. In our case, f is the Boussinesq term, that is to say

f (T,Cl)= ρ[1−βT (T −Tre f )−βC(Cl −Cre f )]g

The solving steps are the same as the ones presented in section (3.4). The differences (high-

lighted in red) appear in the velocity prediction and the pressure correction step. Indeed while

interpolating the pressure at the faces thanks to formula (3.30), we have to be sure to have an

equilibrium between the gradient of pressure and the Boussinesq force at the same time n.

i Velocity prediction

ρuk −un

∆t+div(uk ⊗Qn+1,k−1)= div(µ∇ uk)−∇pn+1,k−1 − µ

Kn+1,k uk + f n+1,k−1 (3.31)

ii Pressure correction ρ

un+1,k−uk

∆t =−∇φk +δ f n+1,k

div(Qn+1,k)= 0(3.32)

with δ f n+1,k = f n+1,k − f n+1,k−1.

Applying the same method than previously we find the following problem.

div(∆t∇φk)= div(ρ uk)+div(∆t δ f n+1,k) (3.33)

iii External forces updatef n+1,k = f n+1,k−1 +δ f n+1,k (3.34)

Remark 9. In practice the equations are not exactly solved as presented in this chapter but an

incremental formulation is used for efficiency purposes. This implementation follows the same

identical steps as the one presented before. The details of this formulation can be found in Appendix

A.

3.6 Treatment of the porous matrix

In the mushy zone the solid and the liquid phases coexist, so that a cell contains liquid and some

solid obstacles. Code_Saturne handles with this porous layer using a porosity formulation. This

formulation has been detailed in [8] and in Code_Saturne this feature is found setting (iporous=3).

Its main purpose is to kill the momentum fluxes inside each solidified part of the alloy. Thus

when gl = 0, we set to 0 the momentum fluxes calculated at the faces of the cell.

27

3.7. MESH GENERATION AND POST-PROCESSES

3.7 Mesh generation and post-processes

We used the Salome platform to create the different meshes needed and to post-process the final

results. It is an open-source software that provides a generic pre and post-processing platform for

numerical simulation.

28

Validation benchmarks4T his chapter is dedicated to the verification and the validation of our model thanks to

well-known benchmarks. The first case was simulated by Voller and Prakash [10] and

is often used to test solidification models. They chose a simple model, not taking into

consideration the solute transport equation and considering a given liquid volume fraction profile.

The results obtained have been compared with the software SOLID which is based on an averaged

equations model presented in Appendix B. The second case was inspired by Hebditch and Hunt

experiments but with a modified geometry and boundary conditions. This benchmark has been

simulated by 5 other industrial and academic codes with whom we compared our results. The

different formulations and implementations of the model presented in chapters 2 and 3 have

been tested on these cases and a summary of their comparisons can be found below. We have also

compared our results with those provided by some experimental data. We replicated first two

well-known experiments of freezing inside a thin thermal cavity performed by Hebditch and Hunt.

Finally we simulated an ingot casting realized by Ascometal, a French steel company.

4.1 Voller and Prakash benchmark

4.1.1 Case description

Voller and Prakash focused on the freezing of a metallurgical alloy in a parallelelipedic thermal

cavity. In [10], they derived the governing equations of the problem inside a classical mixture

theory framework. To solve this problem, they proposed an enthalpy formulation associated with

a fixed grid numerical method. They modeled the effects induced by the mushy region by choosing

suitable source terms in the balance equations. An important feature of the problem is that

the solute transport equation is not considered. So that in this first approach, we focus on the

solidification of an homogeneous mixture. Finally the case is considered as a 2D problem and all

the physical quantities have been non dimensionalized.

4.1.2 Geometry of the problem

The considered geometry is represented below, it is a squared cavity of length 1.

29

4.1. VOLLER AND PRAKASH BENCHMARK

Figure 4.1: Geometry of Voller and Prakash benchmark with 5 monitoring points

In Code_Saturne, we set 5 monitoring points (A, B, C, D, E and F) as output control. They are

represented in Fig (4.1) and their coordinates can be found in the table below. At the end of the

simulation, we will have several data files listing the time evolution of the main variables for

each probes location.

Table 4.1: Monitoring points coordinates

x y zA 0.25 0.5 0.0B 0.5 0.75 0.0C 0.75 0.5 0.0D 0.5 0.25 0.0E 0.05 0.25 0.0F 0.05 0.75 0.0

4.1.3 Characteristics of the problem

We consider a binary alloy whose non dimensionalized properties are listed in Table 4.2. We also

report the initial and boundary conditions of the problem. The phase change takes place over a

30


range of temperature T ∈ [−ε;ε] with ε arbitrarily chosen.

Table 4.2: Test problem data for Voller and Prakash benchmark

Initial temperature T0 = 0.5Hot wall temperature Thot = 0.5Cold wall temperature Tcold =−0.5Reference temperature Tre f = 0.5Half mushy temperature range ε= 0.1Cavity length l = 1Specific heat c = 1Density ρ = 1Viscosity µ= 1Thermal conductivity k = 0.001Coefficient of thermal expansion β= 0.01Latent heat L = 5Liquidus temperature Tl = 0.1Solidus temperature Ts =−0.1

4.1.4 Modeling and numerical setup

4.1.4.1 Mesh

We consider only cartesian grids with a unique cell in the z direction in order to discretize this

2D problem. In Figure (4.2), we can observe the 50x50 mesh used for our simulation, in this case

we have an homogeneous mesh size dx = 0.02.

Figure 4.2: Mesh 50x50 with the 6 monitoring points

31


4.1.4.2 Governing equations

One major simplification of this benchmark is the non-inclusion of the species transport equation

into the modeling set of equations. In this test problem, Voller and Prakash considered only the

conservation of mass, the conservation of momentum and the heat equation. Those equations

have been derived in chapter 2. We gather equations (3.1), (2.42) and (2.46) below.∂t(ρu)+div(ρu⊗u)=−∇(p)+div(µ∇u)+ρre f (1−βT (T −Tre f ))g+F

div(u)= 0

∂t(ρcpT)+div(ρcpTu)= div(λ∇T)−∂t(ρglL)

(4.1)

with F imposed as

F =−1600(1− gl)2

g3l

u

Remark 10. To avoid dividing by zero when the liquid volume fraction is null, we introduce an

arbitrary small quantity q in the source term F.

F =−1600(1− gl)2

g3l + q

u (4.2)

We chose q = 0.001 for our simulations as Voller and Prakash did in [10].

We cannot observe numerical instabilities when putting gl = 0 in equation (4.2), because in this

case we have u = gl ul = 0

Remark 11. As explained in section 2.5, Voller and Prakash used the following enthalpy source

term.

Sh = ∂(ρglL)∂t

+div (ρglLu)

Yet taking into consideration the enthalpy crossed-terms, we proved that the divergence term inside

Sh cancel, see equation (2.46).

In addition, Voller and Prakash imposed the liquid volume fraction as follows.

gl(T)=

1 i f T > Tl iq

T −Tsol

Tl iq −Tsoli f Tsol < T < Tl iq

0 i f T < Tsol

(4.3)

Where Tsol and Tl iq respectively stands for the solidus and liquidus temperature and their

modulus is equal to ε.

32


Figure 4.3: Variation of the liquid volume fraction along the temperature

4.1.4.3 Initial and boundary conditions

Initially the temperature inside the cavity is set to Tinitial = 0.5 which is above the liquidus

temperature. The boundary conditions are two adiabatic upper and lower walls, one cold left wall

Tcold =−0.5 and one hot right wall Thot = Tinitial = 0.5. At time t = 0, we impose a temperature

Tcold below the solidus temperature on the left wall. As time goes by, a solidification process

occurs inside the cavity starting from the cold wall.

4.1.5 TEST 1: Comparison between two algorithmic schemes

We compare the two algorithmic schemes presented in chapter 3, SIMPLEC and PISO-like, for

different values of sub-iterations.

N Algorithm Nb sub-iterations Mesh (dx) Time step Final time CPU time

CASE 1 SIMPLEC - 50x50 (0.02) 0.01 500 2h44CASE 2 PISO-like 10 50x50 (0.02) 0.01 500 5h19CASE 3 PISO-like 40 50x50 (0.02) 0.01 500 7h27

Table 4.3: Numerical setup for the different tests performed

The CPU time increases when increasing the maximal number of PISO sub-iterations, it

reaches 7h27 for a maximum number of sub-iterations set to 40. The future work of this internship

33


will focus on reducing the simulation time that is consequent even for this simple case.

0 100 200 300 400 500Time (s)

0.0

0.1

0.2

0.3

0.4

0.5Temperature at point A

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

0.30

0.35

0.40

0.45

0.50Temperature at point B

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50Temperature at point D

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4Temperature at point F

CASE 1CASE 2CASE 3

Figure 4.4: Temperature profiles at 4 monitoring points

We observe that the temperature is initially at T0 = 0.5 and then decreases while the cavity

is being cooled down.

34


0 100 200 300 400 500Time (s)

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00Liquid volume fraction at point A

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

0.94

0.96

0.98

1.00

1.02

1.04

1.06Liquid volume fraction at point B

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

0.94

0.96

0.98

1.00

1.02

1.04

1.06Liquid volume fraction at point D

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

0.0

0.2

0.4

0.6

0.8

1.0Liquid volume fraction at point F

CASE 1CASE 2CASE 3

Figure 4.5: Liquid volume fraction profiles at 4 monitoring points

The liquid volume fractions are changing for points A and F. First crystals of solid appears at

point A from t = 300 and at t = 500 point A is still in the mushy region. Point F solidify really

fast, the mushy region at this point does not even last 20. The liquid volume fraction at points B

and D remains constant and equal to 1, so they are in a zone that is still liquid after 500.

35


0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016Velocity at point A

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012Velocity at point B

CASE 1CASE 2CASE 3

0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012Velocity at point D

CASE 1CASE 2CASE 3

0 50 100 150 200Time (s)

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007Velocity at point F

CASE 1CASE 2CASE 3

Figure 4.6: Velocity profiles at 4 monitoring points

The velocity at points A and F tends to 0. As we have seen, these two points will be inside the

mushy region at one moment of the simulation. When they are inside the mushy zone, they are

subjected to the damping force F, see equation (4.2). This force modeling the effect of the porosity

has the consequence to reduce the velocity until 0. On the contrary the velocity at points B and D

that are still fluid at the end of the simulation does not tend to 0.

We notice that the plots superpose perfectly regardless of the numerical scheme used.

36


0 10000 20000 30000 40000 50000Number of temporal iterations

0

5

10

15

20

25

30

35

40Number of PISO-like sub-iterations

CASE 1CASE 2CASE 3

Figure 4.7: Number of PISO-like sub-iterations for each temporal iteration

The number of sub-iterations performed reduces quickly. After t = 50, CASE 3 uses only 10

sub-iterations as CASE 2.

In this simple benchmark, we observe no differences of our variables of interest between the

two algorithms. So SIMPLEC algorithm can be used as it provides the same results faster.

4.1.6 Time step sensitivity analysis

For a fixed mesh, we vary the time step.

N Algorithm Nb sub-iterations Mesh (dx) Time step Final time

CASE 1 PISO-like 10 50x50 (0.02) 0.1 500CASE 2 PISO-like 10 50x50 (0.02) 0.01 500CASE 3 PISO-like 10 50x50 (0.02) 0.005 500CASE 4 PISO-like 10 50x50 (0.02) 0.001 500


For the case 1, we observe an unstable solution with absurd values for the velocity. So, we

present the comparison at 4 monitoring points for the other cases 2, 3 and 4.

37


0 100 200 300 400 500Time (s)

0.0

0.1

0.2

0.3

0.4


CASE 2CASE 3CASE 3

0 100 200 300 400 500Time (s)

0.30

0.35

0.40

0.45


CASE 2CASE 3CASE 4

0 100 200 300 400 500Time (s)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45


CASE 2CASE 3CASE 4

0 100 200 300 400 500Time (s)

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3


CASE 2CASE 3CASE 4


38


0 100 200 300 400 500Time (s)

0.65

0.70

0.75

0.80

0.85

0.90

0.95


CASE 2CASE 3CASE 4

0 100 200 300 400 500Time (s)

0.94

0.96

0.98

1.00

1.02

1.04


CASE 2CASE 3CASE 4

0 100 200 300 400 500Time (s)

0.94

0.96

0.98

1.00

1.02

1.04


CASE 2CASE 3CASE 4

0 100 200 300 400 500Time (s)

0.0

0.2

0.4

0.6

0.8


CASE 2CASE 3CASE 4


39


0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014


CASE 2CASE 3CASE 4

0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010


CASE 2CASE 3CASE 4

0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010


CASE 2CASE 3CASE 4

0 50 100 150 200Time (s)

0.000

0.001

0.002

0.003

0.004

0.005

0.006


CASE 2CASE 3CASE 4


Once again we observe that the plots superpose perfectly for all the cases. Except for the

CASE 1, for which we obtain unstable results.

For a fixed mesh of 50x50, decreasing the time step does not impact the final results from

dt = 0.01.

4.1.7 Mesh sensitivity analysis

Algorithm Nb sub-iterations Mesh (dx) Time step Final time CPU time

PISO-like 10 20x20 (0.05) 0.05 500 24minPISO-like 10 50x50 (0.02) 0.01 500 5h19PISO-like 10 75x75 (0.013) 0.01 500 1day 6h15PISO-like 10 100x100 (0.01) 0.01 500 2days 8h37


We notice that the CPU time increases almost proportionally to the mesh size.

40


0 100 200 300 400 500Time (s)

−0.1

0.0

0.1

0.2

0.3

0.4


20x2050x5075x75100x100

0 100 200 300 400 500Time (s)

0.30

0.35

0.40

0.45

0.50


20x2050x5075x75100x100

0 100 200 300 400 500Time (s)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45


20x2050x5075x75100x100

0 100 200 300 400 500Time (s)

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3


20x2050x5075x75100x100


41


0 100 200 300 400 500Time (s)

0.4

0.5

0.6

0.7

0.8

0.9


20x2050x5075x75100x100

0 100 200 300 400 500Time (s)

0.94

0.96

0.98

1.00

1.02

1.04


20x2050x5075x75100x100

0 100 200 300 400 5000.94

0.96

0.98

1.00

1.02

1.04


0 10 20 30 40 50 60 70Time (s)

0.0

0.2

0.4

0.6

0.8


20x2050x5075x75100x100


42


0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016


20x2050x5075x75100x100

0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012


20x2050x5075x75100x100

0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012


20x2050x5075x75100x100

0 10 20 30 40 50 60 70Time (s)

0.000

0.001

0.002

0.003

0.004

0.005

0.006


20x2050x5075x75100x100


The temperature reaches the convergence even with a coarse mesh of 50x50. We observe that

to refine the mesh is more important for the liquid volume fraction and for the velocity.

4.1.8 Numerical results: A comparison with an other CFD software

We chose to simulate our case with a 50x50 mesh and a time step equals to 0.01 according to our

previous sensitivity analysis.

We simulate Voller and Prakash case on the software SOLID and compare the results with our

code.

43


Software Mesh (dx) Time step

Code_Saturne 50x50 (0.02) 0.01SOLID 50x50 (0.02) 0.1

Table 4.6: Numerical setup for the simulations performed on SOLID and Code_Saturne

Code_Saturne SOLID

Figure 4.14: Temperature field with 11 contour lines ranging inside [−0.5; 0.5]

44


Code_Saturne SOLID

Figure 4.15: Liquid volume fraction map with the 2 contour lines gl = 0.1 and gl = 0.9

Code_Saturne SOLID

Figure 4.16: Velocity field

The velocity field of the fluid at t = 500 is a vortex due to the thermal convection experienced

45

4.2. HEBDITCH AND HUNT BENCHMARK

by the solidifying cavity. The arrows gives the direction and the intensity of the velocity. The

maximal velocity norm being of the order of 10−2, we can calculate the Reynolds number of our

problem.

Re = ρulµ

≈ 10−2 (4.4)

We deduce that we are in a laminar regime.

We observe that the contour lines of the temperature and of the liquid volume fraction are not

straight lines close to the liquid phase, due to the vortex.

Besides, we denote a good qualitative comparison between the different results provided by

Code_Saturne and by SOLID.

4.2 Hebditch and Hunt benchmark

In 2007, Combeau and al created the SMACS project to study the development of segregations

during alloy solidification. This project involves five French laboratories: CEMEF, IJL, EM2C,

EPM-SIMAP and TREFLE. In 2012, in the article [3], they gathered and compared their numeri-

cal results for the freezing of a Pb−18%Sn alloy. The geometry and boundary conditions used in

this benchmark are similar to the ones used by Hebditch and Hunt during their experiments

presented in section 4.3. The five laboratories concluded that their computational codes are not

fully predictive yet, due to both modeling and numerical aspects. Using a similar geometry and

initial and boundary conditions, we add our predictions to their contribution.


Combeau et al. considered a 2D cartesian rectangular cavity of length l = 0.06m and width

L = 0.05m.

46


Figure 4.17: Geometry of Hebditch and Hunt benchmark

We set 5 monitoring points A, B, C and D inside the cavity.

Table 4.7: Monitoring points coordinates

x y zA 0.04 0.01 0.0B 0.04 0.05 0.0C 0.025 0.03 0.0D 0.02 0.01 0.0


We consider a Pb−18%Sn alloy whose adimensionalized properties are listed in Table 4.8. We

also report the initial and boundary conditions of the problem.

47


Table 4.8: Test problem data for Hebditch and Hunt benchmark

Phase diagram dataInitial mass fraction w0 (wt%) 18.0Melting temperature T f (C) 327.5Liquidus slope m (C wt%−1) −2.334Partition coefficient kp (−) 0.310Eutectic mass fraction weut (wt%) 61.911Thermal dataSpecific heat cp (J kg−1 C−1 176Thermal conductivity λ (W m−1 C−1 17.9Latent heat L (J kg−1) 37600Other characteristicsDensity ρ (kg m−3) 9250Reference temperature Tre f (C) 285.488Thermal expansion coefficient βT (C−1) 1.16 ·10−4

Solutal expansion coefficient βC (wt%−1) 4.9 ·10−3

Dynamic viscosity µ (kg m−1 s−1) 1.1 ·10−3

Secondary dendrite arm spacing λ2 (µm) 18.5Heat transfer coefficient h (W m−2 C−1) 400External temperature Text (C) 25Calculation parametersInitial temperature T0 (C) 285.488Final time t f (s) 600


4.2.3.1 Governing equations

Unlike the previous case, Hebditch and Hunt benchmark takes into consideration the solute

transport equation. Combining equations (3.1), (2.42), (2.46), (2.47), we obtain the following

system.

∂t(ρu)+div(ρu⊗u)=−∇(p)+div(µ∇u)+ρre f (1−βT (T −Tre f )−βC(C−Cre f ))g− µ

Ku

div(u)= 0

∂t(ρcpT)+div(ρcpTu)= div(λ∇T)−∂t(ρglL)

∂t(ρC)+div(ρCu)= div(ρD∇C)−div(ρ(Cl −C)u)(4.5)

Besides, we used equations (2.38) and (2.39) as closure laws for the liquid volume fraction with

the porosity K verifying the Karman-Cozeny law written in equation (2.33).

4.2.3.2 Initial and boundary conditions

A liquid alloy at the temperature T0 = 775C is poured into the cavity. For the first tests, we used

this value as initial temperature and not the one reported in Table 4.8 in order to have a all

48


liquid alloy at the beginning of the simulation. At t = 0s, the right side of the problem is cooled

by a Fourier type condition while a symmetric condition is imposed on the left wall and the last

walls are adiabatic.

4.2.4 TEST 1: Comparison between two algorithmic schemes

Once again we begin by comparing the two algorithmich schemes SIMPLEC and PISO-like for

different maximal numbers of sub-iterations.

N Algorithm Nb sub-iterations Mesh (dx) Time step Final time CPU time

CASE 1 SIMPLEC - 50x50 (0.001m) 0.001 s 600 s 11h55CASE 2 PISO-like 10 50x50 (0.001m) 0.001 s 600 s 1day 1h20CASE 3 PISO-like 40 50x50 (0.001m) 0.001 s 600 s 1day 1h30


49


0 100 200 300 400 500 600Time (s)

400

450

500

550

600

650

700

750

800Temperature at point A

CASE 1CASE 2CASE 3

0 100 200 300 400 500 600Time (s)

400

450

500

550

600

650

700

750

800Temperature at point B

CASE 1CASE 2CASE 3

0 100 200 300 400 500 600Time (s)

400

450

500

550

600

650

700

750

800Temperature at point C

CASE 1CASE 2CASE 3

0 100 200 300 400 500 600Time (s)

400

450

500

550

600

650

700

750

800Temperature at point D

CASE 1CASE 2CASE 3


50


0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8


CASE 1CASE 2CASE 3

0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8


CASE 1CASE 2CASE 3

0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8

1.0Liquid volume fraction at point C

CASE 1CASE 2CASE 3

0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8


CASE 1CASE 2CASE 3


51


0 50 100 150 200 250 300Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016Velocity norm point A

CASE 1CASE 2CASE 3

0 50 100 150 200 250 300Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012Velocity norm point B

CASE 1CASE 2CASE 3

0 50 100 150 200 250 300Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014Velocity norm point C

CASE 1CASE 2CASE 3

0 50 100 150 200 250 300Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014Velocity norm point D

CASE 1CASE 2CASE 3


52


0 100 200 300 400Time (s)

13

14

15

16

17

18

19

20Concentration at point A

CASE 1CASE 2CASE 3

0 100 200 300 400Time (s)

12

13

14

15

16

17

18

19

20Concentration at point B

CASE 1CASE 2CASE 3

0 100 200 300 400Time (s)

14

15

16

17

18

19

20Concentration at point C

CASE 1CASE 2CASE 3

0 100 200 300 400Time (s)

14

15

16

17

18

19

20Concentration at point D

CASE 1CASE 2CASE 3

Figure 4.21: Concentration profiles at 4 monitoring points

We observe that for this benchmark all the points experience the mushy region during a

period of time longer for points C and D than for points A and B. The velocity tends to 0 because

of the damping term of the momentum equation.

The velocity, the volume fraction and the temperature profiles are almost the same for

any numerical scheme used. We observe differences between the schemes when looking at the

concentration at points B and C.

53


0 100000 200000 300000 400000 500000 600000Number of temporal iterations

0

5

10

15

20

25

30

35

40Number of PISO-like sub-iterations

CASE 1CASE 2CASE 3

Figure 4.22: Number of PISO-like sub-iterations for each temporal iteration

We observe that the maximal number of PISO sub-iterations is performed most of the time in

CASE 2 and CASE 3.

4.2.5 TEST 2: The drift formulation

In this section we test the drift formulation for the numerical treatment of the concentration

source term, see equation ( 2.49). Previously the concentration source term −div(ρ(Cl −C)u) was

treated explicitly in the right hand side of the solute balance.

Numerical treatment Mesh (dx) Time step Final time CPU time

No Drift 50x50 (0.001m) 0.001 s 600 s 1day 1h20Drift 50x50 (0.001m) 0.001 s 600 s 19h08

Table 4.10: Numerical setup for the drift formulation comparison

54


0 100 200 300 400 500 600Time (s)

400

450

500

550

600

650

700

750

800Temperature at point A

DriftNo Drift

0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8


DriftNo Drift

0 50 100 150 200 250 300 350 400Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016Velocity norm at point A

DriftNo Drift

0 100 200 300 400 500 600Time (s)

13

14

15

16

17

18

19Concentration at point A

DriftNo Drift

Figure 4.23: Profiles of the main variables at the monitoring point A

At the 4 monitoring points there are no differences of ||u||, gl ,C and T profiles between the

drift and the no drift formulation. In the figure (4.23), only the profiles at point A have been

gathered. We obtained similar results at the other points. Yet, the difference arises when looking

at the CPU time. We see a gain in time with the drift formulation.

4.2.6 TEST 3: Comparison with and without the modified pressureinterpolation

For fixed mesh 50x50 and time step dt = 0.001s, we compare the influence of the modified

pressure interpolation presented in section (3.5.1). We test its influence on the velocity and the

liquid volume fraction fields.

55


Figure 4.24: Velocity field at t = 100s. Left: With the modified pressure interpolation. Right:Without the modified pressure interpolation.

Figure 4.25: Liquid volume fraction map at t = 100s. Left: With the modified pressure interpola-tion. Right: Without the modified pressure interpolation.

Without the improved pressure interpolation, we detect heterogeneities in the velocity and

the liquid volume fraction fields. On the contrary the improved pressure interpolation smoothen

these maps.

56


4.2.7 TEST 4: Comparison with and without the porosity formulation

The porosity formulation has been presented in section (3.6). Dropping the add-ons related to the

porosity implementation, we obtain the following results for the bulk concentration field.

Figure 4.26: Bulk concentration field obtained the porosity implementation. Left: t = 300s. Right:t = 600s.

We observe that segregated channels appear during the freezing but at the end of solidification

there is no channel left. The solute has been transported by heat fluxes that would have been

annihilated with the porosity implementation.

4.2.8 Time step sensitivity analysis

We first perform a time step sensitivity analysis for a fixed mesh.

Mesh Time step Final time CPU time50x60 (1.0 ·10−3 m) 0.01 s 600 s 7h5750x60 (1.0 ·10−3 m) 0.005 s 600 s 12h50x60 (1.0 ·10−3 m) 0.001 s 600 s 1day50x60 (1.0 ·10−3 m) 0.0005 s 600 s 2.5days

Table 4.11: Numerical setup for the different codes compared in this section

57


CASE 1 CASE 2

CASE 3 CASE 4

Figure 4.27: Segregation maps for different time steps

We observe unphysical results before the CASE 3 with a time step dt = 0.001. Indeed for

CASE 1 and CASE 2, we observe some jumps of the concentration at the bottom and at the upper

part of the map that have no physical meaning.

4.2.9 Mesh sensitivity analysis

We use a fixed time step dt = 0.001 and we refine the mesh.

58


Algorithm Nb sub-iterations Mesh Time step Final time CPU time

PISO-like 10 50x60 (1.0 ·10−3 m) 0.001 s 600 s 1day 4h06PISO-like 10 100x120 (5.0 ·10−4 m) 0.001 s 600 s 3days 16h36PISO-like 10 200x240 (2.5 ·10−4 m) 0.001 s 600 s 19days 15h25

Table 4.12: Numerical setup for the different codes compared in this section

Mesh 50x60 Mesh 100x120 Mesh 200x240

Figure 4.28: Segregation map at the end of the freezing for different meshes.

We observe that refining the mesh gives more precise results on the number and the shape of

the mesosegragtions. Yet the macrosegregations, negative at the bottom and positive at the left

upper corner, are well predicted in intensity and shape from a mesh of 100x120.

Below, we cut our cavity at y = 0.05m and compare the segregation profiles obtained for

different meshes.

59


0.00 0.01 0.02 0.03 0.04 0.05X (m)

0

10

20

30

40

50

60Bulk concentration profile at y=0.05m

50x60100x120200x240

Figure 4.29: Concentration profile plotted over the horizontal line y= 0.05m at t = 600s

For the 3 meshes, peaks of concentration are observed at the same place.

Finally, we plot the temperature, liquid volume fraction, concentration and velocity norm at point

C located at the middle of the geometry.

60


0 100 200 300 400 500 600t (s)

350

400

450

500

550

600Temperature at point C

50x60100x120200x240

0 100 200 300 400 500 600t (s)

0.0

0.2

0.4

0.6

0.8

1.0Liquid volume fraction at point C

50x60100x120200x240

0 100 200 300 400 500 600t (s)

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018Velocity norm at point C

50x60100x120200x240

0 100 200 300 400 500 600t (s)

14

15

16

17

18

19Bulk concentration at point C

50x60100x120200x240

Figure 4.30: Evolution of the temperature, the liquid volume fraction, the concentration and thevelocity norm at point C

The temperature, the liquid volume fraction and the velocity norm are similar from one mesh

to the other. But the solute concentration can endure consequent changes when refining the mesh.

We could refine more to see if we can reach a convergence in a reasonable time.

4.2.10 Numerical results: Comparison with SOLID

We simulated Hebditch and Hunt benchmark on the software SOLID and compared the results

with those provided by our code.

61


0 50 100 150 200 250 300 350 400Time (s)

400

450

500

550

600

650

700

750

800Point A

0 50 100 150 200 250 300 350 400Time (s)

400

450

500

550

600

650

700

750

800Point B

0 50 100 150 200 250 300 350 400Time (s)

450

500

550

600

650

700

750

800Point C

0 50 100 150 200 250 300 350 400Time (s)

450

500

550

600

650

700

750

800Point D

Code SaturneSOLID


62


0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8

1.0Point A

0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8

1.0Point B

0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8

1.0Point C

0 100 200 300 400 500 600Time (s)

0.0

0.2

0.4

0.6

0.8

1.0Point D

Code SaturneSOLID


63


0 50 100 150 200 250Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016Point A

0 50 100 150 200 250Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012Point B

0 100 200 300 400 500Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014Point C

0 50 100 150 200 250 300 350 400Time (s)

0.000

0.002

0.004

0.006

0.008

0.010

0.012Point D

Code SaturneSOLID


64


0 50 100 150 200 250 300 350 400Time (s)

13

14

15

16

17

18

19Point A

0 50 100 150 200 250 300 350 400Time (s)

12

13

14

15

16

17

18

19

Point B

0 50 100 150 200 250 300 350 400Time (s)

14

15

16

17

18

19

Point C

0 50 100 150 200 250 300 350 400Time (s)

14

15

16

17

18

19Point D

Code SaturneSOLID

Figure 4.34: Concentration profiles at 4 monitoring points

We observe that the temperature and the liquid volume fraction provide quite similar results.

The differences for the velocity were expected because the momentum equation solved by the 2

codes are different, see Appendix B. The concentration are in good agreement except for point B

which is more segregated in the SOLID case.

4.2.11 Numerical results: Comparison with 5 other codes

In this section, we compare our result with 5 other numerical codes that used the same volume

averaged model presented in Appendix B. The main difference with the continuum mixture

theory appear in the momentum equation formulation. Plus they used the same set of hypothesis

presented in section (2.4.2). The different computer codes are based on a finite volume (FV) or a

finite element (FE) approach. Their main features combined to the mesh size and time step used

for the simulations are listed in the following table. The last line corresponds to the case we have

65


simulated on Code_Saturne.

Group Software Mesh Time step (s)

IJL SOLID (FV) 192x232 5 ·10−3

CEMEF R2SOL (FE) 46502 nodes 5 ·10−3

EPM-SIMAP FLUENT (FV) 200x240 5 ·10−3

TREFLE THETIS (FV) 268x324 1 ·10−3

IJL OpenFOAM (FV) 200x240 5 ·10−3

EDF R&D Code_Saturne (FV) 200x240 1 ·10−3

Table 4.13: Numerical setup for the different codes compared in this section taken from [9].

We compare the concentration maps obtained after complete solidification at t = 600s.

Figure 4.35: Segregation map after complete solidification obtained with Code_Saturne

66


Figure 4.36: Segregation maps after complete solidification obtained by 5 different codes. Imagetaken from [3]

67


We notice that our final solute concentration field has a similar aspect as the one provided by

the other computational codes. The main differences appear at the level of the segregations. The

segregations predicted by Code_Saturne map quite well with the ones of SOLID.

We also simulated this benchmark on SOLID in order to make a further comparison between

the two codes. Doing so, we could compare our numerical predictions with those provided by

SOLID at different time of the freezing. Thus, we could check that the front interface moves at a

similar velocity in both cases.

Figure 4.37: Concentration field t = 150s. Left: Code_Saturne. Right: SOLID.

68


Figure 4.38: Liquid volume fraction field t = 150s. Left: Code_Saturne. Right: SOLID.

Figure 4.39: Temperature field t = 150s. Left: Code_Saturne. Right: SOLID.

69

4.3. HEBDITCH AND HUNT EXPERIMENT

We observe a good qualitative comparison between our code and SOLID. The differences arise

at the level of the mesosegregations. But the intensity and shapes of the macrosegregations are

similar.

4.3 Hebditch and Hunt experiment

In 1974, Hebdicth and Hunt realized an experience to study the formation of macrosegregations

in ingots. Two alloys Pb-48%Sn and Sn-5%Pb were used to observe the impact of a change

of solute density on macrosegregations. They used a quenching technique, which is the rapid

freezing of the metal in water, in order to measure the components concentrations at different

stage of the solidification. The governing equations solved in this section are exactly the same as

the ones ruling the Hebditch and Hunt benchmark, see equation (4.5)

4.3.1 Experimental design

Hebditch and Hunt used a parallelepipedic cavity of dimensions small enough to have a fast

quench and big enough to observe convective behaviors and to keep a long solidification time.

At first, the homogenized liquid alloy was poured into the mold composed by three adiabatic

walls and it was cooled down from the left side during 1 to 2 hours. 2 to 8 thermocouples were

used to measure the temperature at specific points of the ingot and concentration measures were

performed by spectrophometry. The recorded concentration values were considered to be accurate

to ±2%

70


Figure 4.40: Unidirectional solidification experiment from [5]. A: Alloy specimen. B: Chill block.C: Thermocouples.

To model numerically this experience we use the following geometry and boundary conditions.

71


Figure 4.41: Geometry of the numerical domain


This case is really similar to Hebditch and Hunt benchmark except that the geometry is twice

longer and we do no longer impose a symmetry condition on one side, we instead consider 3

adiabatic walls. We simulate the freezing for two different binary alloys whose properties can

be found in Table 4.14. The alloys have been chosen in such a way that they do not experience

the same direction of solutal convection. For the alloy Sn−5%Pb the solute has a higher density

than the solvent, thus the solutal convective direction is downward. We expect the positive

segregations to form at the bottom. It is exactly the contrary for the alloy Pb−48%Sn.

Figure 4.42: Thermosulatal convection for the two alloys. Image from [9]

72


4.3.3 Numerical results

In the two cases, we used a mesh 100x60 (dx = 1 mm) and a time step dt = 0.001 s.

Results for a Sn−5%Pb alloy. The concentration maps we observe below are in agreement

with our conjecture to see positive segregations at the bottom of the cavity, see figure (4.42).

73


Figure 4.43: C−C0C0

at t = 400s (R2SOL)


at t = 400s (SOLID)


at t = 400s (Code_Saturne)

74


0.00 0.02 0.04 0.06 0.08 0.10Length of the cavity (m)

−2

−1

0

1

2

3

4

5

6C − C0 at y = 5mm

Numerical resultsExperimental measures


−2

−1

0

1

2

3

4

5

6C − C0 at y = 25mm


−2

−1

0

1

2

3

4

5

6C − C0 at y = 35mm


−2

−1

0

1

2

3

4

5

6C − C0 at y = 55mm

Figure 4.46: Gap between the concentration and the initial concentration along different horizon-tal lines

Results for a Pb−48%Sn alloy. The concentration map we observe is in agreement with

our conjecture to see positive segregations in the upper part of the cavity, see figure (4.42).

75



at t = 400s (R2SOL)


at t = 400s (SOLID)


at t = 400s (Code_Saturne)

76

4.4. INDUSTRIAL APPLICATION


−5

0

5

10

15

C − C0 at y = 5mm

Numerical resultsExperimental measures


−5

0

5

10

15

C − C0 at y = 25mm


−5

0

5

10

15

C − C0 at y = 35mm


−5

0

5

10

15

C − C0 at y = 55mm

Figure 4.50: Gap between the concentration and the initial concentration along different horizon-tal lines

4.4 Industrial application

In this section, we will apply our model to an industrial application taking as toy case a 6.2

tons ingot casted by Ascométal. The main difference compared to the previous simulations is the

consideration of the thermal interactions between the ingot mold and the alloy. Indeed the mold

(solid) and the alloy (solid and liquid) are both subjected to their own physical equations, but they

interact at the interface exchanging heat fluxes. We accounted for those interactions using the

77


internal coupling frame inside Code_Saturne.


Following a similar approach as Kumar and al [7], we simplify the geometry of the mold by

considering a simple cylinder with a similar volume than the real mold. In a first approach we

did not considerate the refractories.

Figure 4.51: Geometry of the industrial ingot


For our simulations we used one coarse mesh (dx = 2.5cm) and one more refined mesh (dx = 1cm).

I did not have the time to test a finer mesh (dx = 0.5cm), but it could be interested to further the

mesh sensitivy analysis.

78


Figure 4.52: Coarse mesh (dx = 2.5cm)

The data used to simulate the dynamics inside the alloy and the mold can be found in [7].

79


4.4.3 Numerical results

The entire solidification of the ingot has been reached at t = 7900s ≈ 2h11m for the two meshes

used. We obtain the following final segregation maps.

Figure 4.53: Solute concentration after complete solidification. Left: Coarse mesh (dx = 2.5cm).Right: Refined mesh (dx = 1cm)

80


Our code captured the main features of the macrosegrgeations: a positive segregation at

the top left corner and a negative one at the bottom. It succeeds in predicting the shape and

intensity of the biggest segregations independently of the mesh. Plus with a refined mesh, we

could identified some mesosegrations.

81

Conclusion

During this internship a continuum mixture model has been implemented in Code_Saturne.Considering usual limiting assumptions such as an immobile solid phase, the lever rule as

microsegregation model, the Darcy law to model the porosity, I derived a solidification model

composed by a set of strongly coupled equations following the same steps as in [12]. This model

involves several source terms modeling the interface liquid-solid contributions. I added these

source terms in the dedicated user subroutines of Code_Saturne writing in C or Fortran90. A

PISO-like algorithmic scheme to handle better with the strong coupling of the model has been

chosen. Besides an improved interpolation of the pressure and a better implementation of the

porosity have been used. The PISO-like algorithm, the improved interpolation of the pressure and

the porosity were already implemented in Code_Saturne, but I had to adapt them to our model.

Then I have performed simulations on different benchmarks using SALOME platform to do both

pre-processing and post-processing (definition of the geometry, build of the meshes, statistical

treatment and visualization of the results) and compared the results with diverse numerical

codes and sometimes with available experimental data. Finally I run the code on an industrial

application of a 6.2 tons ingot casted by Ascométal.

The code provides results in good agreement with the other numerical and experimental data.

It succeeds to catch the macrosegregations shape and intensity. Yet, some improvements can

be done to better capture the mesosegregations. Improvements to reduce the CPU time of the

simulations could be carried out.

Future work will focus on:

• Applying the code to more realistic geometries. The geometry of the industrial ingot, simu-

lated in Chapter 4, can be complexified adding some refractories or exothermic powders.

• Comparing the numerical results with an analytical solution.

• Choosing more realistic set of hypothesis. For example, considering a non fixed solid phase

or taking different values of the physical coefficients λ, D in the liquid and solid phases.

• Reducing the simulation time using a listing of the time passed in each steps of the

numerical process.

82

Appendix A: The incremental PISO-like formulation

In Code_Saturne the equations are not exactly solved as presented in Chapter 3. For efficiency

purposes, we introduce instead at the k− th sub-iteration incremental velocities (δu)k and (δδu)k

defined by

(δu)k = uk −un

(δδu)k = uk −un+1,k−1

= (uk −un)+ (un −un+1,k−1)

= (δu)k + (un −un+1,k−1)

We denote by AD the following advection-diffusion operator applied to the velocity vector and to

a scalar ψ:

AD(u)= div(u⊗Q)−div(µ∇u) (6)

AD(ψ)= div(ψQ)−div(λψ∇ψ) (7)

Then (3.31) can be written as

ρ(δu)k

∆t+AD((δu)k)=−∇pn+1,k−1 − µ

Kn+1,k (δu)k − µ

Kn+1,k un + f n+1,k−1 −AD(un) (8)

And using (δδu)k, we obtain the following prediction step

ρ(δδu)k

∆t+AD((δδu)k)=−∇pn+1,k−1 − µ

Kn+1,k (δδu)k − µ

Kn+1,k un+1,k−1 + f n+1,k−1 −AD(un+1,k−1)

(9)

We can also rewrite the correction step (3.32) with−ρ (δδu)k

∆t =−∇φk +δ f n+1,k

div(Qn+1,k)= 0(10)

In a similar way, the heat and solute transport equations are respectively solved with incremental

temperature and concentration unknowns. Let’s defined the incremental temperature (δT)k and

84


concentration (δC)k as follow.

(δT)k = Tn+1,k −Tn

(δC)k = Cn+1,k −Cn

Then the equations (3.12) and (3.13) rewrite as follow

ρ(δC)k

∆t+AD((δC)k)=−AD(Cn)−div((Cn

l − (δC)k)Qn+1,k−1)−div((Cnl −Cn)Qn+1,k−1) (11)

ρcp(δT)k

∆t+AD((δT)k)=−AD(Tn)−

∆(ρgn+1,k/k−1l L)

∆t(12)

85

Appendix B: The averaged equations model

The averaged equations model is widely used. It is a single region model as the continuum

mixture model presented in Chapter 2. A unique system of conservation equations is solved in

the whole domain, divided into 3 parts: liquid, solid and mushy zones. The interactions between

the phases at the interface liquid-solid are considered explicitly.

More explanations on this model can be found in [6]. After using similar limiting assumptions

as in section (2.4.2), Bellet and al. [2] come to the following system .

∂(ρu)∂t

+div(ρ

glu⊗u)=−gl∇(p)+div(µl∇u)+ glρ

b g− µl gl

Ku

div(ρu)= 0

∂(ρh)∂t

+ρcp∇T ·u = div(λ∇T)

∂(ρC)∂t

+∇Cl ·u = 0

(13)

We observe that the main difference with the continuum mixture model lies in the momentum

equation formulation. Indeed the momentum balance has been performed only in the liquid

phase as the solid phase is consider immobile. Thus the liquid volume fraction gl is scaling

many contributions of this momentum balance because it has not been compensated with gs as

presented in Chapter 2. It is the model implemented in the numerical codes SOLID and R2SOL.

86

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